Transcript The Normal Distribution

Problem:
Assume that among diabetics the fasting
blood level of glucose is approximately
normally distributed with a mean of
105mg per 100ml and an SD of 9 mg
per 100 ml. What proportion of diabetics
having fasting blood glucose levels
between 90 and 125 mg per 100 ml ?
NORMAL
DISTRIBUTION
AND ITS
APPL ICATION
INTRODUCTION
Statistically, a population is the set of all
possible values of a variable.
Random selection of objects of the
population makes the variable a random
variable ( it involves chance mechanism)
Example: Let ‘x’ be the weight of a newly
born baby. ‘x’ is a random variable
representing the weight of the baby.
The weight of a particular baby is not
known until he/she is born.
Discrete random variable:
If a random variable can only take values
that are whole numbers, it is called a
discrete random variable.
Example: No. of daily admissions
No. of boys in a family of 5
No. of smokers in a group of 100
persons.
Continuous random variable:
If a random variable can take any value, it
is called a continuous random variable.
Example: Weight, Height, Age & BP.
Continuous Probability Distributions
Continuous distribution has an
infinite number of values between
any two values assumed by the
continuous variable
As with other probability
distributions, the total area under the
curve equals 1
Relative frequency (probability) of
occurrence of values between any
two points on the x-axis is equal to
the total area bounded by the curve,
the x-axis, and perpendicular lines
erected at the two points on the x-
Histogram
F r e q u e n cy
20
10
0
1 1 .5
2 1 .5
3 1 .5
4 1 .5
5 1 .5
6 1 .5
7 1 .5
Age
Figure 1 Histogram of ages of 60 subjects
The Normal or Gaussian distribution is the
most important continuous probability
distribution in statistics.
The term “Gaussian” refers to ‘Carl Freidrich
Gauss’ who develop this distribution.
The word ‘normal’ here does not mean
‘ordinary’ or ‘common’ nor does it mean
‘disease-free’.
It simply means that the distribution
conforms to a certain formula and shape.
Histograms
A kind of bar or line chart


Values on the x-axis (horizontal)
Numbers on the y-axis (vertical)
Normal distribution is defined by a
particular shape


Symmetrical
Bell-shaped
A Perfect Normal
Distribution
Gaussian Distribution
Many biologic variables follow this
pattern

Hemoglobin, Cholesterol, Serum Electrolytes, Blood
pressures, age, weight, height
One can use this information to define
what is normal and what is extreme
In clinical medicine 95% or 2 Standard
deviations around the mean is normal

Clinically, 5% of “normal” individuals
are labeled as extreme/abnormal
 We just accept this and move on.
The Normal Distribution
Characteristics of Normal
Distribution
Symmetrical about mean, 
Mean, median, and mode are equal
Total area under the curve above the xaxis is one square unit
1 standard deviation on both sides of
the mean includes approximately 68%
of the total area
 2 standard deviations includes
approximately 95%
 3 standard deviations includes
approximately 99%
Characteristics of the Normal
Distribution
Normal distribution is completely
determined by the parameters 
and 


Different values of  shift the
distribution along the x-axis
Different values of  determine
degree of flatness or peakedness of
the graph
Applications of Normal
Distribution
Frequently, data are normally
distributed


Essential for some statistical procedures
If not, possible to transform to a more
normal form
Approximations for other distributions
Because of the frequent occurrence of
the normal distribution in nature, much
statistical theory has been developed
for it.
What’s so Great about the
Normal Distribution?
If you know two things, you know
everything about the distribution


Mean
Standard deviation
You know the probability of any
value arising
Standardised Scores
My diastolic blood pressure is 100

So what ?
Normal is 90 (for my age and sex)

Mine is high
 But how much high?
Express it in standardised scores

How many SDs above the mean is
that?
Mean = 90, SD = 4 (my age and sex)
My Score - Mean Score 100-90

 2.5
SD
4
This is a standardised score, or z-score
Can consult tables (or computer)


See how often this high (or higher) score
occur
99.38% of people have lower scores
Standard Scores
The Z score makes it possible, under
some circumstances, to compare scores
that originally had different units of
measurement.
Z score
Allows you to describe a particular score in
terms of where it fits into the overall group of
scores.

Whether it is above or below the average and how
much it is above or below the average.
A standard score that states the position of a
score in relation to the mean of the
distribution, using the standard deviation as
the unit of measurement.

The number of standard deviations a score is
above or below a mean.
Z Score
Suppose you scored a 60 on a numerical test
and a 30 on a verbal test. On which test did
you perform better?

First, we need to know how other people did on
the same tests.
 Suppose that the mean score on the numerical test was
50 and the mean score on the verbal test was 20.
 You scored 10 points above the mean on each test.
 Can you conclude that you did equally well on both
tests?
 You do not know, because you do not know if 10 points
on the numerical test is the same as 10 points on the
verbal test.
Z Score
Suppose you scored a 60 on a numerical test
and a 30 on a verbal test. On which test did
you perform better?

Suppose also that the standard deviation on the
numerical test was 15 and the standard deviation
on the verbal test was 5.
 Now can you determine on which test you did better?
Z Score
Z Score
Z score
To find out how many standard
deviations away from the mean a
particular score is, use the Z formula:
Population:
Z
X 

Sample:
XX
Z
S
Z Score
Z
X 

In relation to the rest of the people
who took the tests, you did better
on the verbal test than the
numerical test.
60  50
Z
 .667
15
30  20
Z
2
5
Properties of Z scores
The standard deviation of any
distribution expressed in Z scores is
always one.

In calculating Z scores, the standard
deviation of the raw scores is the unit of
measurement.
Properties of Z scores
Transforming raw scores to Z scores changes
the mean to 0 and the standard deviation to
1, but it does not change the shape of the
distribution.



For each raw score we first subtract a constant
(the mean) and then divide by a constant (the
standard deviation).
The proportional relation that exists among the
distances between the scores remains the same.
If the shape of the distribution was not normal
before it was transformed, it will not be normal
afterward.
The Standard Normal Curve
The Standard Normal Table
Using the standard normal table, you
can find the area under the curve that
corresponds with certain scores.
The area under the curve is
proportional to the frequency of scores.
The area under the curve gives the
probability of that score occurring.
Standard Normal Table
Reading the Z Table
Finding the proportion
of observations
between the mean and
a score when
 Z = 1.80
Reading the Z Table
Finding the proportion
of observations above a
score when
 Z = 1.80
Reading the Z Table
Finding the proportion
of observations
between a score and
the mean when
 Z = -2.10
Reading the Z Table
Finding the proportion
of observations below a
score when
 Z = -2.10
Z scores and the Normal
Distribution
Can answer a wide variety of questions about
any normal distribution with a known mean
and standard deviation.
Will address how to solve two main types of
normal curve problems:


Finding a proportion given a score.
Finding a score given a proportion.
Finding Proportions
Example: Finding a Proportion
Below the Mean
1.
2. Use C’ Column
3. Z  X  

X  20,   30,   10
20  30
Z
 1
10
Example: Finding a Proportion
Below the Mean
4.
1.
Z
X 

2. Use
Column
X C’
 20
,   30, and  10
3.
20  30
Z
 1
10
15.87 % of customers
Example: Finding a Proportion
Between the Mean and a Score
1.
3.
Z
X 

X  116,   100,   16
116  100
Z
1
16
2. Use B Column
Example: Finding a Proportion
Between the Mean and a Score
4.
1.
Z
X 

2. Use
Column
X B
116
,   100, and  16
3.
Z
116  100
1
16
34.13% of the population
Example: Finding a Proportion Below
a Score
Given that IQ is normally distributed with a mean of 100 and a
standard deviation of 16, what proportion of the population has an IQ
below 124?
1.
3.
Z
X 

X  124,   100,   16
Z
2. 0.50 + B Column
124  100
 1.5
16
Example: Finding a Proportion Below
a Score
Given that IQ is normally distributed with a mean of 100
and a standard deviation of 16, what proportion of the
population has an IQ below 124?
4.
.50 + .4332 = .9332
1.
.9332 of the population
X 
Z

2. 0.50 + B Column
X  124,   100, and  16
3. Z 
124  100
 1 .5
16
Example: Finding a Proportion
Between Two Scores
3. Z  X  
1.

X 1  80, X 2  100,   89,   4
80  89
 2.25
4
100  89

 2.75
4
Z x1 
Z x2
2. B’ + B
Example: Finding a Proportion
Between Two Scores
1.
4. .4878+.4970=.9848
98.48%
z
X 

2.XB’
B, X 2  100,   89, and  4
1 +80
80  89
 2.25
4
100  89

 2.75
4
3. Z x 
1
Z x2
Example: Finding a Proportion
Between Two Scores
Given that scores on the Social Adjustment Scale are
normally distributed with a mean of 50 and a standard
deviation of 10, what proportion of scores are between 30
and 40?
X 
Z

1.
3.

X 1  40, X 2  30,   50,   10
40  50
 1
10
30  50

 2
10
Z x1 
Z x2
30
40
2. Larger C’-Smaller C’
Example: Finding a Proportion
Between Two Scores
Given that scores on the Social Adjustment Scale
4.
are normally distributed with a mean of 50 and a
.1587-.0228 =.1359
standard deviation of 10, what proportion of
scores are between 30 and 40?
1.
z
30
X 

40
X 1  40,C’-Smaller
X 2  30,   C’
50, and  10
2. Larger
40  50
 1
3. Z 
10
30  50
Z
 2
10
Finding Scores
Z 
X 

Z  X  
  Z  X
X    Z
Example: Finding a Score from a
Proportion (or Percentile Rank)
Given that scores on the Social Adjustment Scale are normally
distributed with a mean of 50 and a standard deviation
of 10, what two scores correspond to the middle 95%?
1.
4. X    z
Z  1.96,   50, and  10
X  50  1.96(10)
X  30.4and 69.6
2. 0.025 in C and C’
3.
Example: Finding a Score from a
Proportion (or Percentile Rank)
Given that IQ scores are normally distributed with a mean of 100 and a
standard deviation of 16, what score corresponds to the 95th
percentile?
1.
2. 0.50 + 0.45 (B Column)
3.
4. X    Z
z  1.65,   100, and  16
X  100  1.65(16)
X  126.40
Normal Distributions Go
Wrong
Wrong shape

Non-symmetrical
 Skew

Too fat or too narrow
 Kurtosis
Aberrant values

Outliers
Effects of Non-Normality
Skew

Bias parameter estimates
 E.g. mean
Kurtosis

Doesn’t effect parameter estimates
 Does effect standard errors
Outliers

Depends
Distributions
Bell-Shaped (also
known as
symmetric” or
“normal”)
Skewed:


positively (skewed to
the right) – it tails off
toward larger values
negatively (skewed
to the left) – it tails
Kurtosis
Outliers
20
10
0
Value
Dealing with Outliers
Error
Data entry error
 Correct it

Real value
Difficult
 Delete it

ANY
QUESTIONS